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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.

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The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Mathematical physics. --- Physical mathematics --- Physics --- Invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Mathematics --- Affine space. --- Affine transformation. --- Algebra bundle. --- Algebraic surface. --- Almost complex manifold. --- Automorphism. --- Banach space. --- Clifford algebra. --- Cohomology. --- Cokernel. --- Complex dimension. --- Complex manifold. --- Complex plane. --- Complex projective space. --- Complex vector bundle. --- Complexification (Lie group). --- Computation. --- Configuration space. --- Conjugate transpose. --- Covariant derivative. --- Curvature form. --- Curvature. --- Differentiable manifold. --- Differential topology. --- Dimension (vector space). --- Dirac equation. --- Dirac operator. --- Division algebra. --- Donaldson theory. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic surface. --- Equation. --- Fiber bundle. --- Frenet–Serret formulas. --- Gauge fixing. --- Gauge theory. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Hilbert space. --- Hodge index theorem. --- Homology (mathematics). --- Homotopy. --- Identity (mathematics). --- Implicit function theorem. --- Intersection form (4-manifold). --- Inverse function theorem. --- Isomorphism class. --- K3 surface. --- Kähler manifold. --- Levi-Civita connection. --- Lie algebra. --- Line bundle. --- Linear map. --- Linear space (geometry). --- Linearization. --- Manifold. --- Mathematical induction. --- Moduli space. --- Multiplication theorem. --- Neighbourhood (mathematics). --- One-form. --- Open set. --- Orientability. --- Orthonormal basis. --- Parameter space. --- Parametric equation. --- Parity (mathematics). --- Partial derivative. --- Principal bundle. --- Projection (linear algebra). --- Pullback (category theory). --- Quadratic form. --- Quaternion algebra. --- Quotient space (topology). --- Riemann surface. --- Riemannian manifold. --- Sard's theorem. --- Sign (mathematics). --- Sobolev space. --- Spin group. --- Spin representation. --- Spin structure. --- Spinor field. --- Subgroup. --- Submanifold. --- Surjective function. --- Symplectic geometry. --- Symplectic manifold. --- Tangent bundle. --- Tangent space. --- Tensor product. --- Theorem. --- Three-dimensional space (mathematics). --- Trace (linear algebra). --- Transversality (mathematics). --- Two-form. --- Zariski tangent space.

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The description for this book, Seminar On Minimal Submanifolds. (AM-103), Volume 103, will be forthcoming.

Minimal submanifolds. --- A priori estimate. --- Analytic function. --- Banach space. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Cauchy–Riemann equations. --- Center manifold. --- Closed geodesic. --- Codimension. --- Coefficient. --- Cohomology. --- Compactness theorem. --- Comparison theorem. --- Configuration space. --- Conformal geometry. --- Conformal group. --- Conformal map. --- Continuous function. --- Cross product. --- Curve. --- Degeneracy (mathematics). --- Diffeomorphism. --- Differential form. --- Dirac operator. --- Discrete group. --- Divergence theorem. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation. --- Existence theorem. --- Existential quantification. --- Exterior derivative. --- First variation. --- Free boundary problem. --- Fundamental group. --- Gauss map. --- Geodesic. --- Geometry. --- Group action. --- Hamiltonian mechanics. --- Harmonic function. --- Harmonic map. --- Hausdorff dimension. --- Hausdorff measure. --- Homotopy group. --- Homotopy. --- Hurewicz theorem. --- Hyperbolic 3-manifold. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypersurface. --- Implicit function theorem. --- Infimum and supremum. --- Injective function. --- Inner automorphism. --- Isolated singularity. --- Isometry group. --- Isoperimetric problem. --- Klein bottle. --- Kleinian group. --- Limit set. --- Lipschitz continuity. --- Mapping class group. --- Maxima and minima. --- Maximum principle. --- Minimal surface of revolution. --- Minimal surface. --- Monotonic function. --- Möbius transformation. --- Norm (mathematics). --- Orthonormal basis. --- Parametric surface. --- Periodic function. --- Poincaré conjecture. --- Projection (linear algebra). --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Schwarz reflection principle. --- Second fundamental form. --- Semi-continuity. --- Simply connected space. --- Special case. --- Stein's lemma. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subsequence. --- Support (mathematics). --- Symplectic manifold. --- Tangent space. --- Teichmüller space. --- Theorem. --- Trace (linear algebra). --- Uniformization. --- Uniqueness theorem. --- Variational principle. --- Yamabe problem.

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This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Schrodinger equation. --- Wave mechanics. --- Equation, Schrödinger --- Schrödinger wave equation --- Electrodynamics --- Matrix mechanics --- Mechanics --- Molecular dynamics --- Quantum statistics --- Quantum theory --- Waves --- Differential equations, Partial --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- Schrödinger equation. --- Schrödinger, Équation de. --- Solitons. --- Abelian integral. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation. --- Asymptote. --- Asymptotic analysis. --- Asymptotic distribution. --- Asymptotic expansion. --- Banach algebra. --- Basis (linear algebra). --- Boundary (topology). --- Boundary value problem. --- Bounded operator. --- Calculation. --- Cauchy's integral formula. --- Cauchy's integral theorem. --- Cauchy's theorem (geometry). --- Cauchy–Riemann equations. --- Change of variables. --- Coefficient. --- Complex plane. --- Cramer's rule. --- Degeneracy (mathematics). --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differential equation. --- Differential operator. --- Dirac equation. --- Disjoint union. --- Divisor. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Energy minimization. --- Equation. --- Euler's formula. --- Euler–Lagrange equation. --- Existential quantification. --- Explicit formulae (L-function). --- Fourier transform. --- Fredholm theory. --- Function (mathematics). --- Gauge theory. --- Heteroclinic orbit. --- Hilbert transform. --- Identity matrix. --- Implicit function theorem. --- Implicit function. --- Infimum and supremum. --- Initial value problem. --- Integrable system. --- Integral curve. --- Integral equation. --- Inverse problem. --- Jacobian matrix and determinant. --- Kerr effect. --- Laurent series. --- Limit point. --- Line (geometry). --- Linear equation. --- Linear space (geometry). --- Logarithmic derivative. --- Lp space. --- Minor (linear algebra). --- Monotonic function. --- Neumann series. --- Normalization property (abstract rewriting). --- Numerical integration. --- Ordinary differential equation. --- Orthogonal polynomials. --- Parameter. --- Parametrix. --- Paraxial approximation. --- Parity (mathematics). --- Partial derivative. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Pole (complex analysis). --- Polynomial. --- Probability measure. --- Quadratic differential. --- Quadratic programming. --- Radon–Nikodym theorem. --- Reflection coefficient. --- Riemann surface. --- Simultaneous equations. --- Sobolev space. --- Soliton. --- Special case. --- Taylor series. --- Theorem. --- Theory. --- Trace (linear algebra). --- Upper half-plane. --- Variational method (quantum mechanics). --- Variational principle. --- WKB approximation. --- Schrödinger, Équation de.